Question: We know that $\frac{1}{{{n-1}}}>\frac{1}{{{n}}}>0$ for any $n\ge 2$. Considering this fact, what does the direct comparison test say about $\sum\limits_{n=2}^{\infty }~{\frac{1}{n-1}}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A The series converges. (Choice B) B The series diverges. (Choice C) C The test is inconclusive.
Explanation: $\sum\limits_{n=2}^{\infty }~{\frac{1}{{n}}}\,$ is the harmonic series which is known to diverge. Our given series is term-by-term greater than a divergent series, so it also diverges.